Our main aim of this paper is introduced some new unique common random fixed point theorems of random operators in Hilbert Space by considering a sequence of measurable functions satisfying conditions A or B and C. Our results are motivated from [3, 5, 6, 7, 8]. Mathematics Subject Classification: 54H25, 47H10. Keywords: Separable Hilbert space, Random operators, Common random fixed point, Cauchy sequenc

Rajput, Anil

Shrivastava, Rajesh

Singh, Shiv Kumar

English

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Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications  33 Common Fixed Point Theorems  for  Random Operators in Hilbert Space Rajesh Shrivastava*, Anil Rajput** and Shiv Kumar Singh*** *Govt. Science & Commerce Benazeer  College, Bhopal, India (email: rajeshraju0101@rediffmail.com) ** Chardra Shekhar Azad Govt. P.G. College, Sehore ,India  (email: dranilrajput@hotmail.com) ***Sagar Institute of Science and Technology, Bhopal, India (email:lekh13@yahoo.com)  Abstract Our main aim of this paper is introduced some new unique common random fixed point theorems of random operators in Hilbert Space by considering a sequence of measurable functions satisfying conditions A or B and C. Our results are motivated from [3, 5, 6, 7, 8]. Mathematics Subject Classification:  54H25, 47H10. Keywords: Separable Hilbert space, Random operators, Common random fixed point, Cauchy sequence  1. Introduction and preliminaries: In recent years, the study of random fixed points has attracted much attention; some of the recent literatures in random fixed point may be noted in [1, 2, 3].  In this paper we construct a sequence of measurable function and consider its convergence to the common unique random fixed point of two continuous random operators defined on a non- empty closed subset of a separable Hilbert space. For the purpose of obtaining the random fixed point of the two continuous random operators. We have introduced a rational inequality and used the parallelogram law. Throughout this paper, (Ω, Σ) denotes a measurable space consisting of a set  and sigma algebra ⅀ of subset of Ω.   stands for a separable Hilbert space and C is nonempty closed subset of H. 1.1 Definition: A function f: Ω → 	  is said to be measurable if (B⋂ C) ∈ Σ for every Borel subset B of H. 1.2. Definition: A function F: Ω  	 → 	 is said to be a random operator if F(., x) : Ω →→ 	 is measurable for every x ∈ 	. 1.3. Definition:  A measurable function g: Ω → 	 is said to be a random fixed point of the random operator F: Ω  	 →  C if F(t(g(t))) = g(t)  for all  t ∈ Ω. 1.4. Definition:  A random operator F: Ω  	 → C is said to be continuous if for fixed t ∈ Ω,      F(t, .) :  → C is continuous.  1.5. Theorem: Let  C be a non-empty closed subset of a separable Hilbert space H. Let S and T be two continuous random operators defined on C such that for t ∈ Ω, S(t, .), T(t,.):  	 → C satisfy condition (C). Then S and T have a common unique random fixed point in C. and satisfy the following  condition  −  ≤  −  −  1 +  −  + [   −  +     − ] +  C −                      For each x, y in C, a, b, being positive real number such that 0<  +  < 1/2. Proof : We define a sequence of function {&'} as &) : Ω → 	   is arbitrary measurable function for t∈ Ω and n = 0, 1, 2, 3…   . &'*(t) =   S(t, &'(t))            &'*(0) =   T(t, &'*(t))                                      (1)                                                             If &'(t) =&'*(t) = &'*(0)  for t ∈ Ω, for some n then we see that &'(t) is a random fixed point of S and T. therefore we suppose that no two consecutive terms of sequence {&'}  are equal. Now consider for t ∈ Ω &'*(t) − &'*(0) =   S3t, &'(t)4 − T(t, &'*(t))   ≤ S3t, &'(t)4 − T(t, &'*(t))&'(t) − S(t, &'(t))1 + &'(t) − &'*(t)    +b[&'(t) − S(t, &'(t))  + &'*(t) − T(t, &'*(t)) ]     +6[&'(t) − &'*(t)]      =  789:;<(=) 89:;9(>)) 989:(>)89:;<(>)9*89:(>)89:;<(>)9     +[&'(t)−, &'*(t))  + &'*(t)−, &'*(t)) ] Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications  34 +6[&'(t)−, &'*(t)) ]  = (a + b) 3 &'*(t)4 − &'*(t)) + (b + c)&'(t)−, &'*(t))   ⟹ [1 − ( + )]3 &'*(t)4 − &'*(t)) ≤ ( + 6)&'(t)−, &'*(t))   ⟹ 3 &'*(t)4 − &'*(t)) ≤( + 6)[1 − ( + )] &'(t)−, &'*(t)) ⟹ 3 &'*(t)4 − &'*(t)) ≤ K&'(t)−, &'*(t))        Where C =  [ (D*E)[(7*D)]]<9  ≤  In general ⟹ &'(t)−, &'*(t))  ≤ F &'(t)−, &'(t))  ⟹ &'(t)−, &'*(t))  ≤ F' &)(t)−, &(t))      for all  0 ∈ Ω                                                      (2)     Now, we shall prove that for t ∈ Ω , {&'(t)} is a Cauchy sequence. For this for every positive integer p we have, for t ∈ Ω ⟹ &'(t) − &'*G(t)  =  &'(t) −  &'*(t) + &'*(t) −  … … … + &'*G(t) − &'*G(t)        ≤ &'(t) −  &'*(t)  + &'*(t)−, &'*(t))  + ⋯ … . +&'*G(t) − &'*G(t)      ≤ [F' + F'* + ⋯ + F'*G ]&)(t)−, &(t))           by (2)     = F'[1 + C + F … + FG ]&)(t) −  &(t))       ≤ J:(K) &)(t) −  &(t))          for all  0 ∈ Ω as L → ∞,  &'(t) −  &'*G(t)  → 0 it follows  that for 0 ∈ Ω ,  {&'(t)}  is a Cauchy sequence and  hence is convergent in Hilbert space H. For 0 ∈ Ω, let         {&'(t)} → &(0) as n → ∞                                            (3) Since C is closed, g is a function from C to C. Existence of random fixed point:  For  0 ∈ Ω, g(t) − T(t. g(t)) =     &(0) − &'*(t) + &'*(t) −  … … … T(t. g(t))                                            ≤ 2g(t) − &'*(t) + 2&'*(t) − T(t. g(t)) by parlallelogram law  +  ≤ 2[ +  2]                                         = 2g(t) − &'*(t)+230. &'(t)4 − T(t. g(t))                                  ≤ 2g(t) − &'*(t) + 7R3=.89:(>)4S3>,8(=)4989:(>)R3=.89:(>)49*89:(>)8(=)9+                            2b T&'(t) −St, &2Lt2 +&0−Tt,&02  +2c [&2Lt−&02]                             = 2g(t) − &'*(t) +2&'*(t) − T3t, &(0)41 + &'(t) − &(0)+ 2b[&'(t) − &'*(t) + g(t) − T3t, &(0)4] + 2c [&'*(t) − T3t, &(0)4    As {&'*(t)} and {&'*(t)} are subsequences of {&'(t)} as  L → ∞, {&'*(t)} →  g(t)   and {&'*(t)} →&(0) Therefore,                                  ⟹ g(t) − T(t. g(t)) ≤ 2g(t) − g(t) + 2&(0) − &(0)[1 + g(t) − g(t) ]1 + g(t) − g(t) +2b[g(t) − g(t) + g(t) − T3t, &(0)4]                +2	[g(t) − T3t, &(0)4 ]                                ⟹ [1 − 2]g(t) − T3t, &(0)4 ≤ 0    g(t) − T3t, &(0)4 = 0                       ( as 2b < 1) ⟹ T3t, &(0)4 = g(t)                                         ∀ 0 ∈  Ω                                                                  (4)    In an exactly similar way we can prove that for all t ∈  Ω,              ⟹ S3t, &(0)4 = g(t)                                                                                                       (5) Again if  A: Ω	 →  is a continuous random operator on a non empty subset C of a separable Hilbert space H, then for any  measurable function : Ω → 	,   the function  h(t) = Y(0, (0)) is also measurable [1]  It follows from the construction of {&'}  (by (1))  and the above consideration that{&'}    is a sequence of measurable function. From (3), it follow that g is also a measurable function. This  fact along with (4) and (5)  shows that  g: Ω → 	   is common random fixed point of S and T. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications  35 Uniqueness: Let  ℎ: Ω → 	   be another random fixed point common to  S and T that is for t ∈  Ω. S3t, ℎ(0)4 = h(t) T3t, ℎ(0)4 = h(t) Then for     0 ∈  Ω.                  ⟹     g(t) − ℎ(0) =  S(t, g(t) − T3t, ℎ(0)4                                                      ≤ S3t, g(t)4 − T3t, ℎ(0)4g(t) − S3t, g(t)41 + g(t) − h(t) + [g(t) − S3t, g(t)4 + h(t) − T3t, ℎ(0)4] + c[g(t) − h(t)]       ≤ ℎ(0) − ℎ(0)g(t) − g(t)1 + g(t) − h(t) + [g(t) − g(t) + h(t) − h(t)]  + c[g(t) − h(t)]    ⟹  (1 − c)g(t) − ℎ(0) ≤ 0      ⟹  g(t) =  ℎ(0)  for all t ∈  Ω. This complete proof of theorem.  Reference [1] Choudhary, B. S.: A common unique fixed point theorem for two random operators in Hilbert space, IJMMS 32(3) (2002), 177 – 82. [2] Choudhury, B. S. and Ray, M.: Convergence  of an iteration leading to  a solution of a random operator equation, J, Appl. Math. Stochastic Anal., 12(2) (1999), 161 – 168. [3] Choudhury, B. S. and Upadhay, A.: An iteration leading solution and fixed point of operators, Soochow J. Math. 25(4) (1999), 395 – 400. [4] Choudhury, B. S.: Convergence of random iteration scheme to random fixed point, J, Appl. Math. Stochastic Anal.,  8(2) (1995), 139 – 142.  [5] Fisher, B.:  Common fixed point and constant mapping satisfying a rational inequality, Math Sem Kobe Univ., 6(1978), 29 – 35. [6] Himmelberg, C. J.:  Measurable relations, Fund Math, 87 (1975), 53 – 72. [7] Pagey, S. S.,  Shrivastava, S. and Nair, S.: Common fixed point theorem for rational  inequality in a quasi 2- metric space, J. Pure Math, 22 (2005), 99 – 104. [8] Rhoades, B. E.: Iteration to obtain random solution and fixed points of operators in uniformly convex Banach spaces, Soochow Journal of Mathematics 27(4) (2001),         401 – 404.    This academic article was published by The International Institute for Science, Technology and Education (IISTE).  The IISTE is a pioneer in the Open Access Publishing service based in the U.S. and Europe.  The aim of the institute is Accelerating Global Knowledge Sharing.  More information about the publisher can be found in the IISTE’s homepage:  http://www.iiste.org  CALL FOR JOURNAL PAPERS The IISTE is currently hosting more than 30 peer-reviewed academic journals and collaborating with academic institutions around the world.  There’s no deadline for submission.  Prospective authors of IISTE journals can find the submission instruction on the following page: http://www.iiste.org/journals/   The IISTE editorial team promises to the review and publish all the qualified submissions in a fast manner. All the journals articles are available online to the readers all over the world without financial, legal, or technical barriers other than those inseparable from gaining access to the internet itself. Printed version of the journals is also available upon request of readers and authors.  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Common Fixed Point Theorems  for  Random Operators in Hilbert Space

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Common Fixed Point Theorems for Random Operators in Hilbert Space

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